Doing Sums Anthony G. OFarrell National University of Ireland - - PowerPoint PPT Presentation

Doing Sums

Anthony G. O’Farrell

National University of Ireland Maynooth

University of Washington, Seattle, August 2019

1968: On Pennsylvania Avenue

1975: Gamelin’s VII

1974-5: UCLA-Caltech: Gamelin’s XI

2004: Don+Marianne, Brú na Bóinne:

2004: Monasterboice:

Doing sums Beautiful theorems Good open problems

Sums of Algebras If X is a set and F a field, then F X is an algebra over F, when endowed with pointwise operations. We shall consider F = C and F = R.

Sums of Algebras If X is a set and F a field, then F X is an algebra over F, when endowed with pointwise operations. We shall consider F = C and F = R. F[f ] is the subalgebra generated by 1 and f , i.e. the set of all functions x → a0 + a1 · f (x) + · · · + an · f (x)n where each aj ∈ F.

Sums of Algebras If X is a set and F a field, then F X is an algebra over F, when endowed with pointwise operations. We shall consider F = C and F = R. F[f ] is the subalgebra generated by 1 and f , i.e. the set of all functions x → a0 + a1 · f (x) + · · · + an · f (x)n where each aj ∈ F. F[f , g] is the subalgebra generated by 1, f , and g.

Sums of Algebras If X is a set and F a field, then F X is an algebra over F, when endowed with pointwise operations. We shall consider F = C and F = R. F[f ] is the subalgebra generated by 1 and f , i.e. the set of all functions x → a0 + a1 · f (x) + · · · + an · f (x)n where each aj ∈ F. F[f , g] is the subalgebra generated by 1, f , and g. F[f ] + F[g] is the set of all sums h + k, with h ∈ F[f ] and k ∈ F[g]. And so on.

Theorem (1974)

Let ϕ and ψ be homeomorphisms of C into C having opposite

• degrees. Let Y ⊂ C be compact, C \ Y be connected, and

X = bdy(Y ). Then C[ϕ] + C[ψ] is dense in C(X, C).

Theorem (1974)

Let ϕ and ψ be homeomorphisms of C into C having opposite

• degrees. Let Y ⊂ C be compact, C \ Y be connected, and

X = bdy(Y ). Then C[ϕ] + C[ψ] is dense in C(X, C).

Theorem (Walsh 1929, ‘Walsh-Lebesgue’)

Suppose X ⊂ C is compact, the unbounded component Ω of C \ X is connected, and X = bdy(Ω). Then {u ∈ R[x, y] : ∆u = 0 on R2} is dense in C(X, R). i.e. C[z] + C[¯ z] is dense in C(X, C).

Theorem (Browder-Wermer 1964)

Let ψ : S1 → S1 be a direction-reversing homeomorphism. Then every continuous function on S1 can be uniformly approximated by linear combinations of powers zn, n ≥ 0, and ψn, n ≥ 0, i.e. C[z] + C[ψ] is dense in C(S1, C).

Theorem (+Preskenis, 1984)

Let k ∈ N. Let ϕ and ψ be C k diffeomorphisms of C into C having

• pposite degrees. Let X ⊂ C be compact. Then

C[ϕ, ψ] is dense in C k(X, C). Question: What about mere homeomorphisms?

Garnett

Theorem (+Garnett, 1976)

Let ǫ > 0 and ψ be a C 1+ǫ direction-reversing involution of S1 onto

• S1. Then

C[ϕ] + C[ψ] is dense in C 1(S1, C).

Example (+Garnett, 1976) There exists a direction-reversing involution ψ on S1 of class W 1,1 such that C[z] + C[ψ] is not dense in W 1,1(S1, C). This figure has other uses!

Theorem (1986)

C ∞ maps can increase C ∞ dimension. There exists a C ∞ function from R → R2 that has image of C ∞ dimension 2.

Also answers another question: A := closC(S1)C[z], the disk algebra on the circle. For a homeomorphism ψ of S1 onto S1, let Aψ := A ∩ A ◦ ψ.

Theorem (Browder+Wermer, 1964)

If ψ is a singular homeomorphism of the circle onto itself, then Re Aψ is dense in C(S1/ψ, R). Question: When is Aψ trivial, i.e. consisting only of constants? Wermer asked me whether it might hold when ψ is absolutely-continuous. This is not so, and it remains unclear.

Marshall

Going his way:

Lunchtime conversation: a trip

Theorem

(+Marshall, 1979) Let X ⊂ R2 be compact, and suppose all orbits are closed. Then R[x] + R[y] is dense in C(X, R) if and only if there are no round trips in X.

Theorem

(+Marshall, 1979) Let X ⊂ R2 be compact, and suppose all orbits are closed. Then R[x] + R[y] is dense in C(X, R) if and only if there are no round trips in X. Same for A1 + A2, for any two subalgebras of any C(X) that contain the constants.

Theorem

(+Marshall, 1979) Let X ⊂ R2 be compact, and suppose all orbits are closed. Then R[x] + R[y] is dense in C(X, R) if and only if there are no round trips in X. Same for A1 + A2, for any two subalgebras of any C(X) that contain the constants. Big subject: Trips become lightning bolts. Havinson’s example.

µn,(b1,b2,...) := 1

n (b1 − b2 + b3 − b4 + · · · ± bn).

Theorem

(+Marshall, 1983) Let X ⊂ R2 be compact. Then R[x] + R[y] is dense in C(X, R) if and only if µn,b → 0 weak-star for each lightning-bolt b in X.

Proof.

Focus on an extreme norm 1 annihilator, and apply Birkhoff’s Ergodic Theorem to the right map on the space of bolts.

More than 2 algebras

The difference between sums of two algebras and sums of three, or more. Hilbert’s 13th problem.

• Kolmogoroff. Arnold. Vitushkin.

Yaki Sternfeld

Theorem (Sternfeld, 1985)

Let 2 ≤ n ∈ N, and let X be a compact metric space. Then the following are equivalent: (1) X has topological dimension n. (2) ∃ϕi ∈ C(X, R)(i = 1, . . . 2n + 1) such that each f ∈ C(X) is representable as f (x) =

2n+1

• i=1

gi(ϕi(x)) with each gi ∈ C(R). (3) X is homeomorphic to a compact set Y ∈ R2n+1 such that on Y we have: R[x1] + · · · + R[x2n+1] = C(Y , R).

Example (Sternfeld, 1986): There exists a compact set X ⊂ R3 such that each bounded function f ∈ RX is expressible as as sum f = g1(x) + g2(y) + g3(z), with bounded functions gi ∈ RR, but there exists a continuous function f that cannot be so represented using continuous (or even Borel) functions gi.

The culprit: F2.

Reversibility

THANKS